#6487048

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470402
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1296 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

(1,2)(3,8)(4,6)(5,7)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,23)(20,24)(25,26,32,36)(27,30,35,31)(28,37,39,33)(29,34,38,40);(9,11)(10,14)(12,19)(13,17)(15,23)(16,21)(18,20)(22,24)(25,27,31,26,32,35,30,36)(28,34,38,33,39,40,29,37);(1,3,5,4)(2,6,7,8)(11,17)(14,21)(18,19)(22,23)(25,28,32,39)(26,33,36,37)(27,34,35,40)(29,31,38,30);(9,12,13,20)(10,15,16,24)(11,18,17,19)(14,22,21,23)(25,29,32,38)(26,34,36,40)(27,37,35,33)(28,30,39,31);(1,4,5,3)(2,6,7,8)(10,16)(14,21)(15,24)(22,23)(25,30,32,31)(26,35,36,27)(28,38,39,29)(33,34,37,40)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=g,d^2=hi,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,e^b=ei,d^c=di,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6394244);